Internal problem ID [8247]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 667.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.583 (sec). Leaf size: 83
dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(-b*x)+1)*exp(-2*b*x),y(x), singsol=all)
\[ b x +\frac {b \arctanh \left (\frac {2 y \relax (x ) {\mathrm e}^{-b x}-b}{\sqrt {b^{2}+4 b}}\right )}{\sqrt {b^{2}+4 b}}+\ln \left (y \relax (x ) {\mathrm e}^{-b x}\right )-\frac {\ln \left (-b y \relax (x ) {\mathrm e}^{-b x}+y \relax (x )^{2} {\mathrm e}^{-2 b x}-b \right )}{2}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.666 (sec). Leaf size: 95
DSolve[y'[x] == y[x]^3/(E^(2*b*x)*(1 + y[x]/E^(b*x))),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\log (y(x))}{b}+\frac {-\log \left (b e^{b x} \left (e^{b x}+y(x)\right )-y(x)^2\right )+2 \sqrt {\frac {b}{b+4}} \tanh ^{-1}\left (\frac {\sqrt {\frac {b}{b+4}} \left (2 e^{b x}+y(x)\right )}{y(x)}\right )+2 \log \left (e^{b x}\right )}{2 b}=c_1,y(x)\right ] \]